Let's analyze the given system of equations:
[tex]\[
\begin{array}{rcl}
x + y + z & = & 2 \\
2x + 2y + 2z & = & 4 \\
-3x - 3y - 3z & = & -6
\end{array}
\][/tex]
Step 1: Start with the first equation as it is:
[tex]\[ x + y + z = 2. \][/tex]
Step 2: Simplify the second equation:
[tex]\[ 2x + 2y + 2z = 4. \][/tex]
Divide both sides of the equation by 2:
[tex]\[ x + y + z = 2. \][/tex]
This is identical to the first equation.
Step 3: Simplify the third equation:
[tex]\[ -3x - 3y - 3z = -6. \][/tex]
Divide both sides of the equation by -3:
[tex]\[ x + y + z = 2. \][/tex]
This is also identical to the first equation.
Conclusion: All three equations simplify to the same equation:
[tex]\[ x + y + z = 2. \][/tex]
This means that the system of equations does not provide enough independent conditions to find unique values for [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex]. Instead, it defines a plane in three-dimensional space where any point [tex]\((x, y, z)\)[/tex] that lies on the plane [tex]\( x + y + z = 2 \)[/tex] is a solution.
Therefore, the system has infinitely many solutions. Any combinations of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] that satisfy the equation [tex]\( x + y + z = 2 \)[/tex] are valid.
Hence, the correct answer is:
[tex]\[ \boxed{C. \text{infinite solutions}} \][/tex]
